Library AreaMethod.examples_3
Require Import area_method.
Lemma l3_44: forall O A X P Q U R G r N,
on_line_d P O A r ->
inversion Q P O A ->
is_midpoint U P O ->
inter_lc R O X U N ->
inversion G R O A ->
perp G Q O A.
Proof.
area_method.
Qed.
Theorem example6_9 : forall L M N A B C SS:Point,
parallel A M M C ->
parallel A N N B ->
parallel A SS SS L ->
M<>C ->
N<>B ->
SS<>L ->
inter_ll L B C A SS ->
inter_ll N B A C SS ->
inter_ll M A C B SS ->
A**M/M**C+A**N/N**B = A**SS/SS**L.
Proof.
am_before_field.
field_and_conclude.
field_and_conclude.
intuition.
Qed.
Theorem th6_41_b : forall A B C M N L P K X1 X2:Point,
is_midpoint M A B ->
is_midpoint N A C ->
is_midpoint K B C ->
on_parallel X1 A C M ->
on_parallel X2 K B N ->
inter_ll P X1 A X2 K ->
is_midpoint L A K ->
B<>C ->
parallel L P B C ->
2+2<>0 ->
L<>P ->
L**P / B**C = (1+2)/(2+2).
Proof.
geoInit.
eliminate P;area_method.
Qed.
Lemma th6_232 : forall A B C D P Q,
on_parallel_d D A B C 1 ->
is_midpoint P B D ->
is_midpoint Q A C ->
parallel P Q A B.
Proof.
area_method.
Qed.
Lemma th6_234_1 : forall A B C D O E F,
on_parallel D A B C ->
inter_ll O A C B D ->
on_inter_line_parallel E O B C A B ->
on_inter_line_parallel F O A D A B ->
Col O E F.
Proof.
area_method.
Qed.
Lemma th6_239 : forall A B C D P Q R A1 D1 J,
is_midpoint Q B C ->
is_midpoint P B A ->
is_midpoint R C D ->
inter_ll A1 D Q B R ->
inter_ll D1 A Q C P ->
inter_ll J A A1 D D1 ->
parallel A J A1 J ->
A<>J ->
A1<>J ->
A**J/A1**J=- (2+1).
Proof.
area_method.
Qed.
Theorem th6_258 : forall A B C D E F M N P Q:Point,
on_parallel_d D B E A 1 ->
on_line C A E ->
on_line F B D ->
inter_ll M D C A F ->
inter_ll N E F B C ->
inter_ll P M N A D ->
inter_ll Q M N E B ->
D<>P -> E<>Q ->
parallel A P D P ->
parallel B Q E Q ->
A**P/D**P=B**Q/E**Q.
Proof.
area_method.
Qed.
Lemma exercice2_38_3 : forall Y P Q U V A B,
inter_ll Y P Q U V ->
on_line A U V ->
S4 U P V Q <>0 ->
~Col U V Q ->
S A B Y = (S U B V) * (S A P Q) / S4 U P V Q.
Proof.
area_method.
Qed.
Theorem th6_41 : forall A B C M N L P K:Point,
is_midpoint M A B ->
is_midpoint N A C ->
is_midpoint K B C ->
is_midpoint L A K ->
on_inter_parallel_parallel P A C M K B N ->
B<>C ->
P<>L ->
P<>A ->
P<>K ->
parallel L P B C ->
S A B C <> 0 ->
2+2 <> 0 ->
L**P / B**C = (1+2)/(2+2).
Proof.
geoInit.
eliminate P;
area_method.
Qed.